No, you cannot receive Holy Communion in the Catholic Church before being baptized.
Baptism is the first sacrament of initiation and is a prerequisite for receiving any other sacrament, including Holy Communion. The Catholic Church teaches that baptism is necessary for salvation (though there are exceptions for baptism of blood and baptism of desire) and that it cleanses a person of original sin and incorporates them into the Church, the Body of Christ.
Only those who are baptized Catholics (or those who have been validly baptized in another Christian denomination that the Catholic Church recognizes) and who are in a state of grace (i.e., free from mortal sin) are properly disposed to receive Holy Communion.